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 Iterability of exp(x)-1 bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 08/13/2007, 08:54 PM (This post was last modified: 08/13/2007, 08:55 PM by bo198214.) Now I indeed had a look at Quote:[1] P. L. Walker, A class of functional equations which have entire solutions, Bull. Austral. Math. Soc. 38 (198, no. 3, 351-356 but things become more complicated! Lets starting with his theorem: Quote:Theorem 2. Let $\phi$ be an entire function of the form $\phi(z)=z+\sum_{n=1}^\infty c_n z^{n+1}$, where $c_1>0, c_n\ge 0$ for all $n$, and either (i) $c_2\neq c_1^2$ or (ii) $c_3. Then the sequence $(f_n)$ defined in Theorem 1 converges uniformly on every $\overline{S}(0,M)$ to a function $f$ which is an entire non-constant solution of (2). Where (2) is $f(w+1)=\phi(f(w))$. The also mentioned theorem 1 and sequence $f_n$ does not matter yet. What however really bothers me, that it seems not to be true: Let $\phi(x)=x+x^2$. This is a feasible function for theorem 2, with $c_1=1$ and $c_2=0\neq c_1^2$. Now I looked at the (unqiue) half iterate $\phi^{\circ 1/2}(x)=f(f^{-1}(x)+1/2)$ which should be entire too, for comparison some members of its series: $ x+{\frac {1}{2}}{x}^{2}-{\frac {1}{4}}{x}^{3}+{\frac {1}{4}}{x}^{4}-{ \frac {5}{16}}{x}^{5}+{\frac {27}{64}}{x}^{6}-{\frac {9}{16}}{x}^{7}+{ \frac {171}{256}}{x}^{8}-{\frac {69}{128}}{x}^{9}+O \left( {x}^{10} \right)$ and tested convergence at $x=5$ (an entire function has an infinite radius of convergence, so it should converge for every $x$) and what did I find? Divergence! Quote:0, 5.0, 17.50000000, -13.75000000, 142.5000000, -834.0625000, 5757.734375, -38187.57812, 222737.7149, -830118.7301, -3591005.876, 123831803.7, -1672085945. So it seems this proof is also not reliable... Boy that shakes my trust in professional mathematics. « Next Oldest | Next Newest »

 Messages In This Thread Iterability of exp(x)-1 - by bo198214 - 08/11/2007, 09:33 PM RE: Iterability of exp(x)-1 - by Daniel - 08/11/2007, 09:49 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/11/2007, 11:42 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/12/2007, 09:07 AM RE: Iterability of exp(x)-1 - by jaydfox - 08/12/2007, 04:41 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 08:54 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/11/2007, 10:25 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 09:06 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 09:13 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 09:16 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 09:33 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 10:00 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:05 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 10:11 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:22 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:00 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/13/2007, 10:29 PM RE: Iterability of exp(x)-1 - by andydude - 08/13/2007, 10:30 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/13/2007, 10:39 PM RE: Iterability of exp(x)-1 - by andydude - 08/15/2007, 08:36 AM RE: Iterability of exp(x)-1 - by bo198214 - 08/15/2007, 09:21 AM RE: Iterability of exp(x)-1 - by andydude - 08/15/2007, 08:40 PM RE: Iterability of exp(x)-1 - by jaydfox - 08/15/2007, 08:54 AM RE: Iterability of exp(x)-1 - by jaydfox - 08/15/2007, 08:53 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/15/2007, 09:13 PM RE: Iterability of exp(x)-1 - by bo198214 - 08/20/2007, 04:14 PM RE: Iterability of exp(x)-1 - by andydude - 09/05/2007, 08:15 PM RE: Iterability of exp(x)-1 - by bo198214 - 09/07/2007, 02:45 PM RE: Iterability of exp(x)-1 - by Gottfried - 03/15/2008, 09:13 AM RE: Iterability of exp(x)-1 - by bo198214 - 03/15/2008, 01:14 PM RE: Iterability of exp(x)-1 - by Gottfried - 03/15/2008, 08:25 PM RE: Iterability of exp(x)-1 - by Gottfried - 03/18/2008, 10:14 AM

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